1970 Perego pram wheels need retreading

The stock wheel and tire is ;
wheel - 17 x 6.5 inches
TIRE - 225 60 17 702 mm diameter

If you wanted a 235 60 17 that would be ok but the tire diameter is larger by a small margin.

If you are reducing wheel size with 16 inch wheels in order to fit the 16 inch tires mentioned in your question you better make sure that the smaller diameter wheel will clear the brakes front and rear.

For a 235 tire the wheel width will need to be 6.5 inches minimum.

So if you go with ...............
wheel- 16 x 6.5 or 16 x 7 inch
TIRE - 235 60 16(as in your question)- 688 mm diameter

Provided the 16 inch wheel clears the brakes that would be OK but that tire is slightly smaller in diameter than stock but within an acceptable margin.

Make sure whatever tire you select that the tire load rating is not less than the stock tire and not less than the tire load rating stipulated by General Motors for that vehicle..

I love using both rubber and urethane tires. My old 14" band saw runs for 15 years with rubber tires on it; I changed it when it started cracking with urethane tires that I got from bandsawparts.com

Apply Loctite to the surfaces, particularly to ones that are melting. Do not store these wheels exposing them to the sun. Repacement could cost $500.

https://picclick.com/SERVICE-Retreading-Atec-Casey-Casey-2-Casey-Pro-3G-282408051511.html

This could be a big wild guess on how the tire comes off on the wheel. But I'll try to help you this thing. I don't know if your bandsaw has adjustment bolt above the upper wheel and below the lower wheel, but in case there is , use these bolts to center the band saw. Well, this could also be blade tension adjustment. Tires should be a very tight fit and a struggle to get on.
Wait, was tired glued on? If not, reglue.
Is that an old tire? If yes, tire must be well worn out and time to invest new ones.
I recommend you to use this time urethane tires rather than rubber tires.
Rubber tires works fine but urethane tires last longer and doesn't need gluing. When you purchase a new tire, make sure that it is a couple of inches smaller than the wheel to stretch the tire. It is a good advice to slightly tilt the top of the drive wheels towards you. I've been purchasing urethane tire with www.bandsawparts.com. Got one from them and still up running. Piece of advise also,Before installing the new tires, you double-check the surfaces of the pulleys to make sure the surfaces are clean and true.

Check the tires maybe it's worn out. If it is, change the tires. I recommend to glue the tires whether it's urethane or rubber. You can try this site for replacement band tires: www.bandsawparts.com.

the whole tire would be difficult to replace with a rubber one, but you can put bmx tires on the plastic wheels and that will give much greater traction, but shorten the life of the gearboxes. here is a link to how to do it.

http://www.youtube.com/watch?v=AAySJVt6bys

check the air pressure and make sure all tires have the same psi in them. If they do, take it in for a front end alignment or a 4 wheel alignment.

On my neice's ride on Jeep, I coated the tires in Rubber Undercoating (sold at all Auto part and home improvement stores.) This creates incredibly better traction, looks, and protects the wheels.

I have the same problem for a vintage 1860's Pedigree pram. Did you ever get the tires you needed?

Hi!
On these, only the 2 rear wheels are the gear wheels. You will just need to replace the tires on those with the rubber ones if you choose to go that way.  Good luck!